Verify if the function has limit if x-->1 f(x)=(x^2-4x+3)/(x-1)

To verify if the limit exists, for x = 1, we'll substitute
x by 1 in the expression of the function.

lim f(x) = lim
(x^2-4x+3)/(x-1)

lim (x^2-4x+3)/(x-1) =  (1-4+3)/(1-1) =
0/0

We notice that we've get an indetermination
case.

We could apply 2 methods for solving the
problem.

The first method is to calculate the roots of the
numerator. Since x = 1 has cancelled the numerator, then x = 1 is one of it's 2
roots.

We'll use Viete's relations to determine the other
root.

x1 + x2 = -(-4)/1

1 + x2
= 4

x2 = 4 - 1

x2 =
3

We'll re-write the numerator as a product of linear
factors:

x^2-4x+3 =
(x-1)(x-3)

We'll re-write the
limit:

lim (x-1)(x-3)/(x -
1)

We'll simplify:

lim
(x-1)(x-3)/(x - 1) = lim (x - 3)

We'll substitute x by
1:

lim (x - 3) =
1-3

lim (x^2-4x+3)/(x-1) =
-2